Given that with genetic engineering we can customize organisms as bio-weapons. Which species have the most worrying potential to be weaponized for mass destruction?

Background:

Biological warfare — is the use of biological toxins or infectious agents such as bacteria, viruses, and
fungi with intent to kill or incapacitate humans, animals or plants as
an act of war.

Examples:

E.G. bio-weapon species:Anthrax fungus - was first tested as a biological warfare agent by Unit 731 of
the Japanese Kwantung Army in Manchuria during the 1930s; some of this
testing involved intentional infection of prisoners of war, thousands
of whom died. Anthrax, designated at the time as Agent N, was also
investigated by the Allies in the 1940s.

E.G. bio-weapon vector:Anopheles labranchiae (malarial mosquito) - According to Prof Frank Snowden, a history professor at Yale
University whose book The Conquest of Malaria in Italy draws on
American archives and the diaries of Italian soldiers, the scheme was
orchestrated in the autumn of 1943 by Erich Martini, a medical
entomologist, Nazi Party member and friend of the SS commander
Heinrich Himmler.

And finally, Class C, which are emerging threats that may be dangerous if engineered largely due to their novelty and their potential lethality/transmissability. The CDC uses examples like Nipah virus and hantavirus.

So that's the list of what the CDC thinks is a big deal. You'll note influenza isn't even on there. I'd argue that any given novel influenza strain belongs in Class C, and likely not Class A because, while potentially both transmissible and virulent, a known vaccine exists and simply needs to be formulated.

In terms of properties for an organism, I'd suggest the following, in no particular order (it likely depends on what said organism is meant to do):

Highly virulent: The disease needs to be able to cause human disease, with high rates of both morbidity and mortality.

Highly transmissible: Unless the goal is to cause a "one off" incident, the disease needs to be able to establish a productive chain of human-to-human infections. Humans can't be a dead-end host. Ideally, this needs to be direct human to human transmission, rather than a vector born disease like West Nile.

Stable and amenable to engineering. It needs to be something that's easy to culture and grow in large numbers. This one often gets overlooked somewhat, but despite the fact that its now somewhat commonplace, bioengineering isn't exactly easy.

Not amenable to treatment. The organism needs to not be easily treatable - broad spectrum antibiotics, existing vaccine stock, etc. mean that the normal public health infrastructure can address, or at least mitigate, an attack. For an attack to be truly effective, it needs to swamp existing infrastructure, and one way to do that is to necessitate intensive, unusual care in hospitals and out of the ordinary action on the part of public health officials.

No doubt influenza. The horizontal shift makes it very virulent and only a few mutations allow it to transfer from birds to mammals (as little as two). See here. There was a big push not to publish these as it would be so easy to make recombinantly. However, thanks to quick work, we have determined that our seasonal flu vaccine protects against this variant.

I would put all zoonotic viruses into this. They passed a species barrier, are new to our immune system and there are no vaccines against them. The flu is one example, corona viruses are another (SARS or even newer MERS). With SARS we had a lot of luck that this didn't end in a bigger epidemic.
A good book about these viruses is the book "Outbreak" (hence the title) by David Quammen.

A more abstract idea for a Bioweapon species could be one that targets key organisms required for completing the nitrogen or carbon cycle. This would likely cause the entire biosphere to collapse though.